Suppose $Q$ is a quadratic form on $\mathbb{R}^3$ defined by $$Q(x,y,z)=xy + 2xz -2y^2-2z^2$$
I want to find the matrix representing the bilinear form associated with this quadratic form in the standard basis. To do this I have written the quadratic form in the following way: $$Q(x,y,z)=\begin{pmatrix} x & y & z\end{pmatrix} \begin{pmatrix} 0 & \frac{1}{2} & 1 \\ \frac{1}{2} & -2 & 0 \\ 1 & 0 & -2\end{pmatrix} \begin{pmatrix} x \\ y \\ z\end{pmatrix}$$
Is this correct?